Finite Group DatabasebetaFinite Group Databasebeta| order | 756 |
| isoclinism | isoclinic class of C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C3×C7×C3⋊D3 |
| unambiguous semidirect product | (C2×C3)⋊(C3×D3×C7) |
| Fitting, socle | C2×C33×C7 ⋊ C2 |
| derived | C32 ⋊ C22×C3×C7 |
| Hall | C33×C7 ⋊ C22 |
| Sylow | C33 ⋊ C22×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C7 | C3⋊D3 |
| commutator, cosocle, hypocenter | C32 | C22×C3×C7 |
| Frattini | C1 | G |
| Fitting, socle | C2×C33×C7 | C2 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C3×C7 | |
152 conjugacy classes of subgroups.
| transitive degree | μt(G) = 252 |
252 conjugacy classes — view the full character table.
| maximal subgroup of | C3×C3⋊D3×Dic7, C3×#92676, C3×#92678, … |
| maximal normal subgroup of | C3×C3⋊D3×Dic7, C3×#92676, C3×#92678, … |
| maximal quotient of | C3×C7×#634, C3×C4×C7×C3⋊D3, C3×C7×C4⋊(C3⋊D3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(756, 185);